[/quote]
,
be all positive integers smaller than 
such that 
for all 
is the largest positive integer that divides both 
and 
.
.
such that: ![\[ x = \sqrt{ x - \frac{1}{x} } + \sqrt{ 1 - \frac{1}{x} } \]](http://latex.artofproblemsolving.com/e/c/f/ecf7fb3c2805290f2cb07b7fdd7c73e1c7a55b99.png)
be a convex pentagon such that 
.
is the circumcenter of triangle 
.
be the circumcenter of triangle 
.
passes through the midpoint of segment 
.
such that for all 
the following holds:![\[f(x)+f(y)=f(xy)+f(f(x)+f(y))\]](http://latex.artofproblemsolving.com/a/c/a/aca098172ea7ce5663e9dd0dd10c2f1495630230.png)
.
and 
meet at a point belonging to the diameter 
are the incenter and the 
be the midpoint of arc 
be the foot of the
intersects the circumcircle again at 
. For
on the circumcircle of triangle 
intersect 
. Prove
are concyclic.
,
lie on a line in this order and satisfy 
.
and 
,
is acute, and let 
be its orthocentre. Points 
and 
lie on lines 
and 
,
and 
and 
and 
be the circumcircle of triangle 
be the second intersection of 
and 
be a transversal of 
be intersections of 
with 
be a tangent line of 
is cyclic
be integers such that there are consecutive integers 
satisfy 
.
is a perfect square.
a
Prove that
of these coins arranged in a line from left to right. He repeatedly performs the following operation: if there are exactly 
coins showing 
t
the process starting with the configuration 
would be 
,
be the number of operations before Harry stops. For example, 
and 
.
possible initial configurations 
,
,
of the
,
,
, 
.
such that
Y by Mango247
In the triangle 
denote 
the center of the inscribed circle, 
the midpoint of the side 
, 
is the point of contact of the 
-exscribed circle with the side , 
is the midpoint of the arc 
of the circumscribed circle triangle. Line 
intersects for the second time the circle circumscribed around the triangle 
, at the point 
. Prove that line 
divides the segment 
in half.
(D. Hilko)
denote 
the center of the inscribed circle, 
the midpoint of the side 
, 
is the point of contact of the 
-exscribed circle with the side , 
is the midpoint of the arc 
of the circumscribed circle triangle. Line 
intersects for the second time the circle circumscribed around the triangle 
, at the point 
. Prove that line 
divides the segment 
in half.(D. Hilko)
This post has been edited 1 time. Last edited by parmenides51, Jun 26, 2021, 1:22 PM
Reason: incircle to A-excircle, see #3
Reason: incircle to A-excircle, see #3
be the 
be the reflection of 
.
bisects 
is the 
because 
is tangent to 
and 
is also this symmedian by isogonality, so 
as desired.
